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Properties

Label 230.2.l.a.17.5
Level $230$
Weight $2$
Character 230.17
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,2,Mod(7,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 17.5
Character \(\chi\) \(=\) 230.17
Dual form 230.2.l.a.203.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.936950 + 0.349464i) q^{2} +(1.92611 - 1.05174i) q^{3} +(0.755750 - 0.654861i) q^{4} +(2.14017 - 0.647831i) q^{5} +(-1.43713 + 1.65853i) q^{6} +(0.903621 + 1.20710i) q^{7} +(-0.479249 + 0.877679i) q^{8} +(0.981834 - 1.52776i) q^{9} +(-1.77883 + 1.35490i) q^{10} +(-1.39727 + 0.638111i) q^{11} +(0.766917 - 2.05618i) q^{12} +(-2.14621 - 1.60663i) q^{13} +(-1.26848 - 0.815205i) q^{14} +(3.44085 - 3.49869i) q^{15} +(0.142315 - 0.989821i) q^{16} +(0.100061 + 1.39903i) q^{17} +(-0.386031 + 1.77455i) q^{18} +(-2.78544 - 3.21457i) q^{19} +(1.19319 - 1.89111i) q^{20} +(3.01002 + 1.37463i) q^{21} +(1.08617 - 1.08617i) q^{22} +(-0.240201 + 4.78981i) q^{23} +2.19455i q^{24} +(4.16063 - 2.77293i) q^{25} +(2.57235 + 0.755311i) q^{26} +(-0.185355 + 2.59161i) q^{27} +(1.47339 + 0.320516i) q^{28} +(-3.52921 - 3.05808i) q^{29} +(-2.00124 + 4.48055i) q^{30} +(4.99090 - 1.46546i) q^{31} +(0.212565 + 0.977147i) q^{32} +(-2.02017 + 2.69863i) q^{33} +(-0.582665 - 1.27586i) q^{34} +(2.71589 + 1.99799i) q^{35} +(-0.258452 - 1.79757i) q^{36} +(0.241444 - 0.0525229i) q^{37} +(3.73320 + 2.03848i) q^{38} +(-5.82360 - 0.837307i) q^{39} +(-0.457085 + 2.18885i) q^{40} +(-3.09304 + 1.98778i) q^{41} +(-3.30062 - 0.236065i) q^{42} +(0.792220 + 1.45084i) q^{43} +(-0.638111 + 1.39727i) q^{44} +(1.11156 - 3.90573i) q^{45} +(-1.44881 - 4.57176i) q^{46} +(0.417418 + 0.417418i) q^{47} +(-0.766917 - 2.05618i) q^{48} +(1.33158 - 4.53494i) q^{49} +(-2.92926 + 4.05209i) q^{50} +(1.66414 + 2.58946i) q^{51} +(-2.67412 + 0.191257i) q^{52} +(-9.06981 + 6.78958i) q^{53} +(-0.732005 - 2.49298i) q^{54} +(-2.57700 + 2.27086i) q^{55} +(-1.49250 + 0.214589i) q^{56} +(-8.74595 - 3.26207i) q^{57} +(4.37539 + 1.63193i) q^{58} +(-14.1593 + 2.03580i) q^{59} +(0.309270 - 4.89741i) q^{60} +(-2.71282 - 9.23902i) q^{61} +(-4.16410 + 3.11721i) q^{62} +(2.73136 - 0.195351i) q^{63} +(-0.540641 - 0.841254i) q^{64} +(-5.63408 - 2.04808i) q^{65} +(0.949723 - 3.23446i) q^{66} +(3.79423 + 10.1727i) q^{67} +(0.991794 + 0.991794i) q^{68} +(4.57497 + 9.47834i) q^{69} +(-3.24288 - 0.922911i) q^{70} +(4.07344 - 8.91960i) q^{71} +(0.870343 + 1.59391i) q^{72} +(-7.82356 - 0.559552i) q^{73} +(-0.207866 + 0.133587i) q^{74} +(5.09744 - 9.71687i) q^{75} +(-4.21019 - 0.605334i) q^{76} +(-2.03286 - 1.11003i) q^{77} +(5.74902 - 1.25062i) q^{78} +(0.466190 + 3.24242i) q^{79} +(-0.336660 - 2.21058i) q^{80} +(4.63192 + 10.1425i) q^{81} +(2.20337 - 2.94335i) q^{82} +(3.47072 + 15.9546i) q^{83} +(3.17501 - 0.932268i) q^{84} +(1.12049 + 2.92935i) q^{85} +(-1.24929 - 1.08251i) q^{86} +(-10.0140 - 2.17840i) q^{87} +(0.109583 - 1.53217i) q^{88} +(14.6523 + 4.30230i) q^{89} +(0.323442 + 4.04792i) q^{90} -4.04247i q^{91} +(2.95513 + 3.77720i) q^{92} +(8.07176 - 8.07176i) q^{93} +(-0.536973 - 0.245227i) q^{94} +(-8.04381 - 5.07522i) q^{95} +(1.43713 + 1.65853i) q^{96} +(3.26012 - 14.9865i) q^{97} +(0.337177 + 4.71435i) q^{98} +(-0.397003 + 2.76121i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{7}{22}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.936950 + 0.349464i −0.662524 + 0.247108i
\(3\) 1.92611 − 1.05174i 1.11204 − 0.607220i 0.185210 − 0.982699i \(-0.440703\pi\)
0.926831 + 0.375479i \(0.122522\pi\)
\(4\) 0.755750 − 0.654861i 0.377875 − 0.327430i
\(5\) 2.14017 − 0.647831i 0.957112 − 0.289719i
\(6\) −1.43713 + 1.65853i −0.586704 + 0.677093i
\(7\) 0.903621 + 1.20710i 0.341536 + 0.456239i 0.938119 − 0.346312i \(-0.112566\pi\)
−0.596583 + 0.802551i \(0.703475\pi\)
\(8\) −0.479249 + 0.877679i −0.169440 + 0.310306i
\(9\) 0.981834 − 1.52776i 0.327278 − 0.509255i
\(10\) −1.77883 + 1.35490i −0.562517 + 0.428456i
\(11\) −1.39727 + 0.638111i −0.421292 + 0.192398i −0.614769 − 0.788707i \(-0.710751\pi\)
0.193477 + 0.981105i \(0.438024\pi\)
\(12\) 0.766917 − 2.05618i 0.221390 − 0.593569i
\(13\) −2.14621 − 1.60663i −0.595252 − 0.445600i 0.258696 − 0.965959i \(-0.416707\pi\)
−0.853948 + 0.520359i \(0.825798\pi\)
\(14\) −1.26848 − 0.815205i −0.339017 − 0.217873i
\(15\) 3.44085 − 3.49869i 0.888424 − 0.903357i
\(16\) 0.142315 − 0.989821i 0.0355787 − 0.247455i
\(17\) 0.100061 + 1.39903i 0.0242683 + 0.339316i 0.995052 + 0.0993582i \(0.0316790\pi\)
−0.970783 + 0.239958i \(0.922866\pi\)
\(18\) −0.386031 + 1.77455i −0.0909883 + 0.418266i
\(19\) −2.78544 − 3.21457i −0.639024 − 0.737473i 0.340178 − 0.940361i \(-0.389513\pi\)
−0.979202 + 0.202888i \(0.934967\pi\)
\(20\) 1.19319 − 1.89111i 0.266806 − 0.422865i
\(21\) 3.01002 + 1.37463i 0.656840 + 0.299969i
\(22\) 1.08617 − 1.08617i 0.231573 − 0.231573i
\(23\) −0.240201 + 4.78981i −0.0500855 + 0.998745i
\(24\) 2.19455i 0.447961i
\(25\) 4.16063 − 2.77293i 0.832126 − 0.554587i
\(26\) 2.57235 + 0.755311i 0.504480 + 0.148129i
\(27\) −0.185355 + 2.59161i −0.0356716 + 0.498755i
\(28\) 1.47339 + 0.320516i 0.278445 + 0.0605719i
\(29\) −3.52921 − 3.05808i −0.655359 − 0.567872i 0.262425 − 0.964952i \(-0.415478\pi\)
−0.917784 + 0.397081i \(0.870023\pi\)
\(30\) −2.00124 + 4.48055i −0.365375 + 0.818033i
\(31\) 4.99090 − 1.46546i 0.896393 − 0.263205i 0.199090 − 0.979981i \(-0.436201\pi\)
0.697303 + 0.716776i \(0.254383\pi\)
\(32\) 0.212565 + 0.977147i 0.0375766 + 0.172737i
\(33\) −2.02017 + 2.69863i −0.351666 + 0.469771i
\(34\) −0.582665 − 1.27586i −0.0999262 − 0.218808i
\(35\) 2.71589 + 1.99799i 0.459070 + 0.337722i
\(36\) −0.258452 − 1.79757i −0.0430753 − 0.299595i
\(37\) 0.241444 − 0.0525229i 0.0396931 − 0.00863471i −0.192675 − 0.981263i \(-0.561716\pi\)
0.232368 + 0.972628i \(0.425353\pi\)
\(38\) 3.73320 + 2.03848i 0.605604 + 0.330685i
\(39\) −5.82360 − 0.837307i −0.932522 − 0.134076i
\(40\) −0.457085 + 2.18885i −0.0722714 + 0.346088i
\(41\) −3.09304 + 1.98778i −0.483052 + 0.310439i −0.759406 − 0.650617i \(-0.774510\pi\)
0.276354 + 0.961056i \(0.410874\pi\)
\(42\) −3.30062 − 0.236065i −0.509297 − 0.0364256i
\(43\) 0.792220 + 1.45084i 0.120812 + 0.221251i 0.931085 − 0.364801i \(-0.118863\pi\)
−0.810273 + 0.586053i \(0.800681\pi\)
\(44\) −0.638111 + 1.39727i −0.0961988 + 0.210646i
\(45\) 1.11156 − 3.90573i 0.165701 − 0.582232i
\(46\) −1.44881 − 4.57176i −0.213616 − 0.674069i
\(47\) 0.417418 + 0.417418i 0.0608867 + 0.0608867i 0.736895 − 0.676008i \(-0.236291\pi\)
−0.676008 + 0.736895i \(0.736291\pi\)
\(48\) −0.766917 − 2.05618i −0.110695 − 0.296785i
\(49\) 1.33158 − 4.53494i 0.190225 − 0.647848i
\(50\) −2.92926 + 4.05209i −0.414260 + 0.573052i
\(51\) 1.66414 + 2.58946i 0.233027 + 0.362597i
\(52\) −2.67412 + 0.191257i −0.370833 + 0.0265225i
\(53\) −9.06981 + 6.78958i −1.24583 + 0.932620i −0.999333 − 0.0365223i \(-0.988372\pi\)
−0.246502 + 0.969142i \(0.579281\pi\)
\(54\) −0.732005 − 2.49298i −0.0996132 − 0.339251i
\(55\) −2.57700 + 2.27086i −0.347482 + 0.306202i
\(56\) −1.49250 + 0.214589i −0.199444 + 0.0286757i
\(57\) −8.74595 − 3.26207i −1.15843 − 0.432072i
\(58\) 4.37539 + 1.63193i 0.574516 + 0.214284i
\(59\) −14.1593 + 2.03580i −1.84338 + 0.265038i −0.973578 − 0.228357i \(-0.926665\pi\)
−0.869805 + 0.493395i \(0.835756\pi\)
\(60\) 0.309270 − 4.89741i 0.0399265 − 0.632253i
\(61\) −2.71282 − 9.23902i −0.347341 − 1.18294i −0.929179 − 0.369631i \(-0.879484\pi\)
0.581838 − 0.813305i \(-0.302334\pi\)
\(62\) −4.16410 + 3.11721i −0.528841 + 0.395886i
\(63\) 2.73136 − 0.195351i 0.344119 − 0.0246119i
\(64\) −0.540641 − 0.841254i −0.0675801 − 0.105157i
\(65\) −5.63408 − 2.04808i −0.698821 − 0.254033i
\(66\) 0.949723 − 3.23446i 0.116903 − 0.398134i
\(67\) 3.79423 + 10.1727i 0.463539 + 1.24279i 0.933011 + 0.359848i \(0.117171\pi\)
−0.469472 + 0.882947i \(0.655556\pi\)
\(68\) 0.991794 + 0.991794i 0.120273 + 0.120273i
\(69\) 4.57497 + 9.47834i 0.550761 + 1.14106i
\(70\) −3.24288 − 0.922911i −0.387599 − 0.110309i
\(71\) 4.07344 − 8.91960i 0.483429 − 1.05856i −0.498078 − 0.867132i \(-0.665961\pi\)
0.981506 − 0.191429i \(-0.0613121\pi\)
\(72\) 0.870343 + 1.59391i 0.102571 + 0.187845i
\(73\) −7.82356 − 0.559552i −0.915678 − 0.0654906i −0.394459 − 0.918913i \(-0.629068\pi\)
−0.521219 + 0.853423i \(0.674523\pi\)
\(74\) −0.207866 + 0.133587i −0.0241639 + 0.0155292i
\(75\) 5.09744 − 9.71687i 0.588602 − 1.12201i
\(76\) −4.21019 − 0.605334i −0.482942 − 0.0694366i
\(77\) −2.03286 − 1.11003i −0.231666 − 0.126499i
\(78\) 5.74902 − 1.25062i 0.650949 − 0.141605i
\(79\) 0.466190 + 3.24242i 0.0524505 + 0.364801i 0.999095 + 0.0425230i \(0.0135396\pi\)
−0.946645 + 0.322278i \(0.895551\pi\)
\(80\) −0.336660 − 2.21058i −0.0376397 − 0.247150i
\(81\) 4.63192 + 10.1425i 0.514658 + 1.12694i
\(82\) 2.20337 − 2.94335i 0.243321 − 0.325039i
\(83\) 3.47072 + 15.9546i 0.380961 + 1.75125i 0.624184 + 0.781278i \(0.285432\pi\)
−0.243223 + 0.969970i \(0.578205\pi\)
\(84\) 3.17501 − 0.932268i 0.346422 − 0.101719i
\(85\) 1.12049 + 2.92935i 0.121534 + 0.317732i
\(86\) −1.24929 − 1.08251i −0.134714 − 0.116730i
\(87\) −10.0140 − 2.17840i −1.07361 − 0.233549i
\(88\) 0.109583 − 1.53217i 0.0116816 − 0.163330i
\(89\) 14.6523 + 4.30230i 1.55314 + 0.456043i 0.942037 − 0.335508i \(-0.108908\pi\)
0.611103 + 0.791551i \(0.290726\pi\)
\(90\) 0.323442 + 4.04792i 0.0340938 + 0.426689i
\(91\) − 4.04247i − 0.423766i
\(92\) 2.95513 + 3.77720i 0.308093 + 0.393800i
\(93\) 8.07176 − 8.07176i 0.837003 − 0.837003i
\(94\) −0.536973 − 0.245227i −0.0553845 − 0.0252933i
\(95\) −8.04381 − 5.07522i −0.825278 − 0.520707i
\(96\) 1.43713 + 1.65853i 0.146676 + 0.169273i
\(97\) 3.26012 − 14.9865i 0.331015 − 1.52165i −0.447649 − 0.894209i \(-0.647739\pi\)
0.778664 − 0.627441i \(-0.215898\pi\)
\(98\) 0.337177 + 4.71435i 0.0340600 + 0.476221i
\(99\) −0.397003 + 2.76121i −0.0399003 + 0.277513i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 230.2.l.a.17.5 ✓ 240
5.3 odd 4 inner 230.2.l.a.63.11 yes 240
23.19 odd 22 inner 230.2.l.a.157.11 yes 240
115.88 even 44 inner 230.2.l.a.203.5 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.5 ✓ 240 1.1 even 1 trivial
230.2.l.a.63.11 yes 240 5.3 odd 4 inner
230.2.l.a.157.11 yes 240 23.19 odd 22 inner
230.2.l.a.203.5 yes 240 115.88 even 44 inner